An effective scheme for solving a class of nonlinear doubly singular boundary value problems through quasilinearization approach
type: Item
instance of: scholarly article
title: An effective scheme for solving a class of nonlinear doubly singular boundary value problems through quasilinearization approach
zbMATH Open document ID: 1468.34029
author: Yanyan Li
published in: Journal of Computational and Applied Mathematics
publication date: 2021-04-23T00:00:00Z
review text: This paper offers an analytical method for solving doubly singular boundary value problems for the equation \[(x^\alpha y')'=x^\beta f(x,y), \;x\in(0,1),\eqno(1)\] with boundary conditions \[y'(0)=0,\quad ay(1)+by'(1)=c\eqno(2)\] or \[y(0)=A,\quad ay(1)+by'(1)=c.\eqno(3)\] Here \(a>0\), \(b\geq0\), \(A, c, \alpha\) and \(\beta\) are constants with \(\beta+2>\alpha\), \({\beta>-1}\), \({\alpha\geq0}\) for (1),(2), \(0\leq\alpha<1\) for (1),(3), and the function \(f(x,y)\) is continuous on \([0,1]\times\mathbb{R}\) and has continuous bounded second derivative with respect to \(y\) on a suitable set. The nonlinear problem (1),(2) is reduced to the following sequence of linear problems \[(x^\alpha y'_{r+1})'=x^\beta f_y(x,y_r)(y_{r+1}-y_r)+x^\beta f(x,y_r), \] \[y'_{r+1}(0)=0,\quad ay_{r+1}(1)+by'_{r+1}(1)=c,\] where the first approximation is \(y_0(x)=\frac{c}{a}.\) Next, using the integral representation of each linear boundary value problem and applying the Picard method, the authors get the folowing combined iterative scheme \[y^{(p+1)}_{r+1}(x)=\frac{c}{a}-\frac{b}{a}\int_0^1\Bigl(t^\beta f_y(t,y_r)(y^{(p)}_{r+1}-y_r)+t^\beta f(t,y_r)\Bigr)\,dt\] \[-\int_0^1G(x,t)\Bigl(t^\beta f_y(t,y_r)(y^{(p)}_{r+1}-y_r)+t^\beta f(t,y_r)\Bigr)\,dt, \] where \(G(x,t)\) is the Green function depending on \(\alpha\), \(y_1^{(0)}(x)=y_0(x)=\frac{c}{a}\) and \(y^{(0)}_{r+1}(x)=y^{(p)}_r(x).\) The convergence of \(\{y^{(p+1)}_{r+1}\}\) to \(y_{r+1}(x)\) and of \(\{y_{r+1}\}\) to the solution \(y(x)\) of (1),(2) is studied. A similar process is organized for the boundary value problem (1),(3). Several examples are given.
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34A45
zbMATH DE Number: 7337441
zbMATH Keywords: nonlinear singular boundary value problem
zbMATH Keywords: quasilinearization
zbMATH Keywords: Picard's method
zbMATH Keywords: approximate analytical solutions
reviewed by: Petio S. Kelevedjiev
MaRDI profile type: Publication
full work available at URL: j.cam.2021.113411
OpenAlex ID: W3128081278
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DOI: 10.1016/J.CAM.2021.113411
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description: scientific article; zbMATH DE number 7337441